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Abstract:

Method for mapping a 3D grid or mesh from a faulted subsurface domain to
a continuous design domain, wherein the grid may be used to represent a
discrete model of a subsurface material property (such as permeability)
to use, for example, in a reservoir simulator. The mapping is
geometry-based, not physics-based. The mapping is determined by an
iterative optimization procedure designed to penalize deformation of
tessellated mesh cells (703) in the design domain compared to their
geometric quality in the faulted domain (701), but subject to stitching
constraints (702) appearing as a penalty term or Lagrange multiplier term
in the optimization objective function to influence the final mesh to
co-locate pairs of points identified on opposite sides of a fault as
having been located together before the fault occurred.

Claims:

1. A method for generating a model of a material property of a faulted
subsurface region for hydrocarbon prospecting or reservoir development,
said method comprising: generating, using a computer, a mapping of a
model mesh representing a physical domain of the subsurface region, with
faults, to an optimized mesh representing a continuous design space in
which all faults are removed, said mapping being designed to minimize
deformation in mesh cells; assigning values of the material property to
continuous volumes in the optimized mesh to generate a model of the
material property in the design space, and using that to generate a model
of the material property in the faulted physical domain; and using the
model of the material property in the faulted physical domain for
hydrocarbon prospecting or reservoir development in the subsurface
region.

2. The method of claim 1, wherein minimizing deformation in mesh cells
comprises: generating a tessellated mesh dividing the physical domain
into cells, and recording geometric quality of each cell; designing
stitching constraints to stitch together discontinuities at fault
boundaries, or alternatively stitching discontinuities by node
relocation, thereby truncating cells at fault boundaries; and optimizing
the mesh in an iterative optimization procedure, subject to the stitching
constraints, with the optimization aimed at minimizing degradation in
geometric quality from the recorded geometric quality due to the
stitching constraints, wherein all mesh nodes are free to move, or all
mesh nodes are free to move except mesh nodes associated with the
stitching together of discontinuities at fault boundaries, which mesh
nodes are relocated to an average position and held fixed there.

3. The method of claim 2, wherein the mesh is optimized to adjust
locations of mesh nodes under influence of the constraints.

4. The method of claim 2, wherein the optimization procedure penalizes
worst quality cells based on a global grid quality measure computed by
adding together quality metrics computed on cells in the mesh.

5. The method of claim 4, wherein the quality metrics computed on every
cell in the mesh are based on combining a shape quality indicator with a
size metric.

6. The method of claim 5, wherein the shape quality indicator is based on
a Jacobian of a mapping from a unit square to a general quadrilateral
cell.

7. The method of claim 2, wherein all cells in the generated tessellated
mesh have edges that do not cross horizon or fault surfaces.

8. The method of claim 2, wherein stitching discontinuities at fault
boundaries includes stitching boundary points on a surface of
discontinuity.

9. The method of claim 8, wherein stitching discontinuities at fault
boundaries further includes parameterizing the surface of discontinuity
and stitching pairs of points intermediate between said boundary points.

10. The method of claim 2, wherein the stitching constraints are based on
minimizing distance between two points, on opposite sides of a fault
boundary, to be stitched together, said two points having been determined
to be co-located before the fault occurred.

11. The method of claim 10, wherein the stitching constraints are imposed
by including, in a cost or objective function that is being minimized in
the optimization, a term containing an expression for said distance
between two points to be stitched together, expressed with a Lagrange
multiplier or as a penalty term.

12. The method of claim 2, wherein the generated tessellated mesh is cut
or non-conforming across discontinuities.

Description:

CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of U.S. Provisional Patent
Application 62/081,159 filed Nov. 18, 2014 entitled "HANDLING DOMAIN
DISCONTINUITY WITH THE HELP OF GRID OPTIMIZATION TECHNIQUES", the
entirety of which is incorporated by reference herein.

FIELD OF THE INVENTION

[0002] This disclosure relates generally to the field of hydrocarbon
operations, such as prospecting or reservoir management and, more
particularly, to reservoir modeling and simulation. Specifically, the
disclosure relates to a method for optimal construction of a conceptual
three-dimensional (3D) grid that is adapted to a subsurface domain's
discontinuities, where the grid may be used for reservoir simulation
studies in reservoir exploration, development or production stages, as
well as for representing a geologic model description of a reservoir
structure and material properties. More specifically, the grid can carry
a model of material properties, such as rock and fluid properties, of a
reservoir or can be used for numerical discretization of partial
differential equations, such as fluid flow or wave propagation.

BACKGROUND

[0003] Consider the general problem of populating a complex n-dimensional
(nD) domain with material properties where the domain is comprised of
multiple separate nD pieces (volumes). A domain is a defined volume
within a space. The pieces may come in partial contact with each other,
thus, forming a non-manifold topology. The domain's material properties
are described by a "designer" who can assign them to only one continuous
volume at a time. To assist the designer's work, the original domain in
the physical space can be mapped to a "design space" where all the
separate volumes are pieced together based on some geometric criterion.
The goal is to construct this mapping in such a way that volumetric
pieces (e.g., compartments that are delineated by horizons and faults)
are minimally deformed and the design space, while being a continuous
volume, still preserves resemblance to the original domain, thus
facilitating the designer's work of populating it with material
properties.

[0004] For example, in geologic modeling of the subsurface, a 3D model
domain is delineated by horizons and faults, where horizons are mostly
flat horizontal surfaces related to deposition of sediment material
forming a reservoir rock, and faults are discontinuities in the rock
introduced by non-depositional events. The rock properties are usually
described by the modeler in a continuous "depositional" space, while the
physical space of the model may contain volume discontinuities in the
form of post-depositional faults. Construction of design space
corresponds to generation of a continuous region from a faulted
structural framework by removing the fault throws.

[0005] U.S. Pat. No. 7,480,205 to Wei ("3D fast fault restoration")
describes a method to solve geo-mechanical equations for a displacement
field using a mesh that conforms to the horizons and faults in the
framework. This is a complex, physics-based computation that is sensitive
to the quality of the supporting mesh and can have performance (speed)
limitations.

[0006] In U.S. Patent Application Publication No. 2008/0021684, a
"parametric" mapping to design space is defined by solving a constrained
optimization problem for three transfer functions u,v,t on a supporting
3D tetrahedral mesh that conforms to fault surfaces. Only tetrahedral
mesh can be used, some of the constraints are heuristic and may be
case-dependent, and special handling is required for erosional horizons.

[0007] Other conventional approaches, such as U.S. Pat. No. 6,106,561, are
based on utilizing the ijk indexing system of the corner point grid built
in the physical space for mapping to "simulation box" design space. Thus,
generation of the mapping logic is combined with the logic for
corner-point grid generation. Such kinds of mappings are very approximate
and do not account for volume distortion of corner-point cells.

[0008] Accordingly, there remains a need in the industry for apparatus,
methods, and systems that are more efficient and may be constructed to
lessen problems with discontinuities associated with grid optimization
techniques. The present techniques provide a method and apparatus that
overcome one or more of the deficiencies discussed above.

SUMMARY

[0009] In one or more embodiments, a method for generating a model of a
material property of a faulted subsurface region for hydrocarbon
prospecting or reservoir development is described. The method comprising:
generating, using a computer, a mapping of a model mesh representing a
physical domain of the subsurface region, with faults, to an optimized
mesh representing a continuous design space in which all faults are
removed, said mapping being designed to minimize deformation in mesh
cells; assigning values of the material property to continuous volumes in
the optimized mesh to generate a model of the material property in the
design space, and using that to generate a model of the material property
in the faulted physical domain; and using the model of the material
property in the faulted physical domain for hydrocarbon prospecting or
reservoir development in the subsurface region.

BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Due to patent law restrictions on the use of color, FIGS. 4, 5 and
6 are black-and-white reproductions of color drawings.

[0011] The advantages of the present invention are better understood by
referring to the following detailed description and the attached
drawings, in which:

[0012] FIG. 1 is a schematic diagram illustrating mapping M of a domain
with discontinuities into a continuous "design space" where a "designer"
defines material properties F;

[0013] FIG. 2 illustrates stitching discontinuities on a surface of a
fault F, matching horizon patches A to A' and B to B', and a surface
patch of F between A and B is matched to the corresponding patch of F
between A' and B';

[0015] FIG. 4 illustrates an example of a prismatic mesh of a faulted
subsurface domain;

[0016] FIG. 5 illustrates two examples of surfaces of a subsurface domain
that delineate volumes (horizons and faults), indicating correspondence
of discontinuities on horizons along fault surfaces (fault traces);

[0017] FIG. 6 shows two examples of a prismatic mesh before (left) and
after (right) optimization by the method of the present techniques;

[0018] FIG. 7 is a flow chart showing basic steps in one embodiment of the
method of the present techniques; and

[0019] FIG. 8 is a block diagram of a computer system that may be used to
perform any of the methods disclosed herein.

DETAILED DESCRIPTION

[0020] In the following detailed description section, the specific
embodiments of the present disclosure are described in connection with
preferred embodiments. However, to the extent that the following
description is specific to a particular embodiment or a particular use of
the present disclosure, this is intended to be for exemplary purposes
only and simply provides a description of the exemplary embodiments.
Accordingly, the disclosure is not limited to the specific embodiments
described below, but rather, it includes all alternatives, modifications,
and equivalents falling within the true spirit and scope of the appended
claims.

[0021] Various terms as used herein are defined below. To the extent a
term used in a claim is not defined below, it should be given the
broadest definition persons in the pertinent art have given that term as
reflected in at least one printed publication or issued patent.

[0022] The articles "the", "a" and "an" are not necessarily limited to
mean only one, but rather are inclusive and open ended so as to include,
optionally, multiple such elements.

[0023] As used herein, the term "hydrocarbons" are generally defined as
molecules formed primarily of carbon and hydrogen atoms such as oil and
natural gas. Hydrocarbons may also include other elements or compounds,
such as, but not limited to, halogens, metallic elements, nitrogen,
oxygen, sulfur, hydrogen sulfide (H2S) and carbon dioxide
(CO2). Hydrocarbons may be produced from hydrocarbon reservoirs
through wells penetrating a hydrocarbon containing formation.
Hydrocarbons derived from a hydrocarbon reservoir may include, but are
not limited to, petroleum, kerogen, bitumen, pyrobitumen, asphaltenes,
tars, oils, natural gas, or combinations thereof. Hydrocarbons may be
located within or adjacent to mineral matrices within the earth, termed
reservoirs. Matrices may include, but are not limited to, sedimentary
rock, sands, silicilytes, carbonates, diatomites, and other porous media.

[0024] As used herein, "hydrocarbon exploration" refers to any activity
associated with determining the locations of hydrocarbons in subsurface
regions. Hydrocarbon exploration normally refers to any activity
conducted to obtain measurements through acquisition of measured data
associated with the subsurface formation and the associated modeling of
the data to identify potential locations of hydrocarbon accumulations.
Accordingly, hydrocarbon exploration includes acquiring measurement data,
modeling of the measurement data to form subsurface models and
determining the likely locations for hydrocarbon reservoirs within the
subsurface. The acquired measurement data may include seismic, gravity,
magnetic, electromagnetic and the like.

[0025] As used herein, "hydrocarbon operations", which may be referred to
as which include hydrocarbon prospecting or reservoir management, refers
to any activity associated with hydrocarbon exploration and/or
hydrocarbon production.

[0026] As used herein, "hydrocarbon production" refers to any activity
associated with extracting hydrocarbons from a well or other opening.
Hydrocarbon production normally refers to any activity conducted in or on
the well after the well is completed. Accordingly, hydrocarbon production
or extraction includes not only primary hydrocarbon extraction but also
secondary and tertiary production techniques, such as injection of gas or
liquid for increasing drive pressure, mobilizing the hydrocarbon or
treating by, for example chemicals or hydraulic fracturing the wellbore
to promote increased flow, well servicing, well logging, and other well
and wellbore treatments.

[0027] The present techniques include a method that can utilize any
finite-element mesh with edges that do not cross horizon or fault
surfaces (can be conformal to the surface or can have two different
approximations of the surface from either side of the surface). A mapping
to a design space is found as a result of solving a grid optimization
problem, which is a simpler formulation than a physics-based restoration
problem, is fully defined based on the geometry and topology of the
supporting mesh (no special treatment for different types of surfaces),
and minimizes volumetric distortion of the mapping. The mapping can be
used to build a subsurface physical property model, which can then be
used for hydrocarbon prospecting or reservoir development. One example is
patent application U.S. Ser. No. 62/031,097, entitled "Volumetric grid
generation in a domain with heterogeneous material properties," which is
incorporated by reference herein in all jurisdictions that allow it.
Another related application is U.S. Patent Application Publication No.
2013/0246031, entitled "Methods to Handle Discontinuity in Constructing
Design Space Using Moving Least Squares," which is also incorporated by
reference herein in all jurisdictions that allow it.

[0028] In one embodiment, the present techniques includes a method for
generating a model of a material property of a faulted subsurface region
for hydrocarbon prospecting or reservoir development, said method
comprising: (a) generating, using a computer, a mapping of a model mesh
representing a physical domain of the subsurface region, with faults, to
an optimized mesh representing a continuous design space in which all
faults are removed, said mapping being designed to minimize deformation
in mesh cells; (b) assigning values of the material property to
continuous volumes in the optimized mesh to generate a model of the
material property in the design space, and using that to generate a model
of the material property in the faulted physical domain; and (c) using
the model of the material property in the faulted physical domain for
hydrocarbon prospecting or reservoir development in the subsurface
region.

[0029] Beneficially, the present techniques address the problem of "design
space" construction as a mesh optimization problem. The mapping of a
point P to or from design space is fully defined from the coordinates of
mesh vertices before (physical space) and after optimization (design
space), the location of point P in the mesh (which cell it belongs to),
and the local geometric basis of that cell (e.g., piece-wise linear
interpolation between the vertices of a tetrahedron).

[0030] In this approach, the mathematical formulation of the problem of
computing the mapping is not physics-based, but geometry-based. It
strives to minimize deformation of each individual mesh cell while
imposing constraints on the discontinuities (they need to be "stitched"
together). Thus, minimal distortion in the volumetric domain pieces is
achieved without being constrained by physics-based deformation or
restoration rules.

[0031] In a physics-based approach, it is necessary to establish boundary
conditions that bring domain pieces together. That requires identifying
correspondence of the boundaries, followed by imposing a physical
condition, e.g. introduction of springs that insures minimal energy at
the configuration when the boundaries are stitched. In the geometric
approach disclosed herein, only geometric correspondence of the
boundaries is utilized. Stitching them together becomes a part of grid
optimization formulation through constraint or penalty terms on an
optimization problem. Therefore, the combination of establishing
correspondence between the boundaries of discontinuities and imposing
optimization constraints may be referred to as "stitching" those
discontinuities.

[0032] One aspect of the present techniques involves defining the
stitching approach (e.g., boundary correspondence of discontinuous
volumes in terms of geometric constraints on the grid optimization
problem). The present techniques may be further understood with reference
to FIGS. 1 to 8, which are described further below.

[0033] FIG. 1 is a schematic diagram 100 illustrating a mapping M of
objects from the physical space 104 to the design space 106. In this
diagram 100, various objects, such as objects 110, 112 and 114 form a
discontinuous volume in the physical space 104. The mapping M is utilized
to form a continuous volume of objects 110', 112' and 114' in the design
space 106, which are associated with the objects 110, 112 and 114 in the
physical space 104. As part of this mapping, the point 116 having
coordinates (x, y, z) in the physical space may be mapped to the point
116' in the design space. In the design space, material properties may be
defined for the objects 110', 112' and 114'. These material properties
may be defined by a user. The material properties may include
permeability, porosity, and density.

[0034] In one embodiment, basic steps in the present techniques for
generation of a mapping from the original model domain to its design
space may be summarized, with reference to the flow chart of FIG. 7, as
follows.

[0035] Step 701: Generate tessellation of the model, and record the
geometric quality of each mesh element. For example, each mesh element in
a 3-D grid may be a tetrahedron, and this geometrical classification is
considered the quality of the mesh element or cell for purposes of this
disclosure. A variety of known techniques exist for tetrahedral mesh
generation in a volumetric domain (e.g., Delaunay tetrahedrization, or
advancing front methods). Any of them can be used in the present
techniques. However, the present techniques are not limited to
tetrahedral grids--any finite-element grid, even hybrid of several
element types such as prisms and tetrahedral, can be utilized. To
facilitate the mapping process, it is preferable that the initial grid be
cut or non-conforming across discontinuities (which can always be
achieved by mirroring grid faces on the discontinuities if the generated
grid was conforming)

[0036] Step 702: Stich discontinuities at corresponding fault boundaries,
and determine constraints. The stitch discontinuities may be performed by
establishing correspondence of the boundaries of the surface patches
(n-1) dimension (D) representing the discontinuities, and construct
constraints. The boundaries of discontinuities are straightforward to
determine and match (need to match two discretized (n-2) D
patches--usually means matching two discretized curves which can be done
by matching their independent parameterizations). In the subsurface
modeling application, this is matching fault traces of each horizon
(intersections between fault surface and horizon surface patches) from
both sides of the fault.

[0037] After the boundaries are matched, the entire discontinuity surface
may be matched approximately. However, if the surface of discontinuity is
too curved, stitching only its boundaries may not be enough to achieve
good match of the entire surface. In this case, "parameterization" of the
discontinuity patches can be introduced, and stitching can be defined for
each parametric line, thus forcing surface match along those lines. As an
example, FIG. 2 is a diagram 200 of stitching discontinuities on a
surface of a fault F, matching horizon patches A to A' and B to B', and a
surface patch of F between A and B is matched to the corresponding patch
of F' between A' and B'. FIG. 3A is a diagram 300 of stitching
discontinuities by boundaries only (A-A', B-B'), while FIG. 3B is a
diagram 320 of stitching with internal constraints and parameterization
(C-C' and D-D'). The dashed curve is the actual discontinuity surface.
Introducing two parametric lines C and D, and stitching also along those
lines produces a better match to the actual discontinuity surface.

[0038] Correspondence of the boundaries leads to formulation of the
constraints for the grid optimization problem. Lagrange multiplier
formulation may be used to achieve exact stitching. As an option (e.g.,
for the case where the scale of discontinuities is significant relative
to the model size), optimization can be started with a penalty term for
constraints, followed by a switch to Lagrange multipliers at the end.
Alternatively, the process can begin by relocating nodes on
discontinuities to actually stitch them, followed by the use of
multipliers during optimization to keep them stitched. The constraints
may not be applied until step 703, but constraints may be constructed as
soon as the points to be stitched together are identified, which is in
step 702.

[0039] Step 703: Optimize mesh to restore original geometric quality for
each mesh element, working against constraints from step 702. This may
involve performing a mesh optimization procedure aiming at restoring
original geometric quality for each cell in the mesh, and impose the
constraints formulation of the previous step. For example, one can use
grid optimization techniques in step 701, where a global grid quality
measure may be computed by adding together quality metrics computed on
every cell in the mesh. The global cell quality measure can have an
adjustment to penalize worst quality cells (inside-out or zero-volume
cells). This global quality measure may be optimized under free movement
of mesh nodes. An individual cell quality metric may be chosen to combine
a shape quality indicator with a size metric using a weight
0≦θ≦1. The cell quality metric can be represented in
a universal way for all types of cells through the Jacobian S of the
mapping from a canonical finite-element shape (e.g., a mapping from a
unit square to a general quadrilateral cell). Furthermore, the use of the
Jacobians allows superimposing multiple mappings and, thus, using a
target cell shape and a size definition different from a canonical shape.
In the present case, individual cell Jacobians H and quality metrics of
the original mesh of step 701 may be computed, and then cell quality
metrics can be written in n-dimensions as

H is a Jacobian for mapping to the target cell (n×n matrix), S is a
Jacobian for the current cell (n×n matrix), 0 is a weight in
combination of shape and size metrics, tr is the trace of a matrix
defined as a sum of its diagonal entries, e.g., for n=3, a 3×3
matrix

matrix = s 11 s 12 s 13 s
21 s 22 s 23 s 31 s 32
s 33 , ##EQU00002##

and trS=s11+s22+s33, det is the determinant of a matrix, and
detS=s11(s22s33-s23s32)-s12(s21s33-s23s31)+s13(s21s32-s22s31).

[0040] The problem of optimization of the global mesh quality measure is
then posed as a problem for finding coordinates of all nodes in the mesh
RT=(X1T, . . . , XnT) that satisfy

where q(cell) are quadrature locations inside each cell,
σq(cell)≧0 are corresponding quadrature weights
satisfying Σq(cell)=1Mσq(cell)=1, and
Jacobians S at those locations can be expressed as functions of mesh node
coordinates R. In the present case, without imposing any constraints, a
solution to the optimization problem above is the original mesh of a
discontinuous domain. Thus, changes in the mesh may be achieved through
introduction of stitching constraints that force mesh nodes to
redistribute.

[0041] The method outlined in FIG. 7 handles all discontinuities at once,
so there is no need to develop a mechanism for ordering discontinuities
as long as their correspondence is established. This method effectively
minimizes deformation of the individual mesh elements from their original
state under the global influence of mesh connectivity. The resulting mesh
describes the continuous design space, and the mapping can be evaluated
both ways as piece-wise continuous on the mesh elements. (Note that
resulting mesh is not necessarily conformable at the stitches, but this
has no implication on evaluation of the two-way mapping. If preferred,
post-processing can be performed to make the resulting mesh conformable
by subdividing elements' neighboring stitches.

[0042] In general, it may be expected that imposing constraints of step
702 may introduce high deformations to the mesh elements near
discontinuities and can lead to inverted (negative volume) elements.
Thus, the grid optimization method in step 703 has to be capable of mesh
untangling as well as of optimizing each element quality based on the
targets, and lend itself to a constrained formulation.

[0043] There are several ways constraints can be imposed on the grid
optimization method described in step 703). For example, three different
ways that step 703 can be performed are as follows:

[0044] Constrained Optimization using Lagrange Multipliers

[0045] A constraint may be defined for each pair of points (i1, i2) that
were identified in step 702 as the two points that should be stitched
together and be co-located to remove the discontinuity, where the
constraint may be expressed in the form
gi(R)=Xk(i1)-Xk(i2), k=1, . . . , n. The grid optimization
problem now transforms into minimization of the Lagrangian functional
where all Nconstr constraints (with weights λi) are added
to the original grid quality measure:

[0046] The numerical solution procedure is similar to solution strategy
for the original minimization problem as described in reference Branets,
which is incorporated by reference herein in all jurisdictions that allow
it. See e.g., L. Branets, "A variational grid optimization method based
on a local cell quality metric," PhD Thesis, University of Texas at
Austin (2005). In this thesis, the method is not applied to unfaulting,
and the thesis does not disclose certain features that are disclosed
herein, including: the technique for stitching discontinuities at fault
boundaries and constructing constraints for the optimization.

[0047] Adding Constraints as Penalty Terms

[0048] A constraint may be defined for each pair of points (i1, i2) that
were identified in step 702 as two points that should be stitched
together and be co-located to remove the discontinuity, where the
constraint may be expressed in the form
gi(R)=1/2(Xk(i1)-Xk(i2))2, k=1, . . . , n. The grid
quality measure now has an extra penalty term for the constraints, where
ε is a small number related to the geometric tolerance at which
two points can be considered the same, as noted in the following
equation:

The numerical solution procedure is very similar to solution strategy for
the unconstrained minimization problem. Optionally, this workflow can be
followed by Lagrange multiplier approach to achieve even tighter
stitching.

[0049] Node Relocation

[0050] In this alternative, the original mesh of the discontinuous domain
may be modified by relocating both nodes in each pair of points (i1, i2)
that were identified in step 702 as the two points that should be
stitched together to their average position X(i1),

X ( i 2 ) → X ( i 1 ) + X (
i 2 ) 2 . ##EQU00006##

Then, either those nodes may be maintained as fixed and the unconstrained
grid optimization is run to restore grid cell shapes, or all nodes remain
flexible and either of the two constrained approaches above may be used
to keep them together during constrained grid optimization. The nodes are
grid or mesh locations, while points are locations within the space.

[0051] Note that for convenience, in the description, above it was assumed
that both of the points to be stitched together are actual mesh nodes
(where a node is an intersection of two or more cell edges). However, all
the formulas and algorithms easily generalize to the case where those
stitching points belong to faces of the mesh cells and their coordinates
can be expressed as linear combination of several mesh nodes, which may
be used when additional "parameterization" is introduced in step 702).

[0052] Example Application: Geologic Modeling

[0053] In an example application to geologic modeling, the present
techniques can be applied as follows:

[0054] (1) Build a general
finite-element mesh in the volumes of the physical space of the model
which are delineated by faults and horizons, and record geometric quality
of each cell. This is illustrated in FIG. 4 where a prismatic mesh has
been chosen. FIG. 4 is a diagram 400 of an exemplary prismatic mesh 402
of a faulted subsurface domain. The diagram 400 has three volumetric
pieces 404, 406 and 408, which form a discontinuous volume.

[0055] (2)
Establish correspondence between fault traces of the same horizon and
stitch them together by one or more of the following types of
constraints: a) penalty+Lagrange multipliers, b) relocating mesh nodes
from both traces to the average location for the trace+Lagrange
multipliers, c) Lagrange multipliers only. FIG. 5 is a diagram 500 of two
horizon views 510 and 520 and shows different examples in which the
correspondence of the top horizon's discontinuities due to a fault are
indicated, as shown by fault surface 512 and 522. These are two different
views 510 and 520 of the model, which consists of three fault blocks: a
large central block 408, a triangular-shaped block 406, and a
rectangular-shaped block 404. Two faults are separating each of the
smaller fault blocks from the central one. FIG. 5 at top shows the
triangular fault block, which can be seen to be separated from central
block 508 by two different faults; on the bottom, FIG. 5 shows the
rectangular fault block, which is also separated from the central block
by two different faults. The darker shaded portions in FIG. 5 are the
bottom horizon of the model.

[0056] (3) Run a mesh optimization procedure
aiming at restoring original geometric quality for each cell, imposing
the constraints found in the previous step. FIG. 6 is a diagram 600 of
horizon views 602, 604, 612 and 614. In the horizon views 602, 604, 612
and 614, the two examples of a prismatic mesh before optimization by the
present techniques, as shown in horizon views 602 and 612, and after
optimization by the present techniques, as shown in horizon views 604 and
614. Accordingly, the adjustments from the horizon views 602 and 612 to
the horizon views 604 and 614 is provided by the mapping from the
faulted, physical domain to a continuous design space.

[0057] The enhanced subsurface model from the present techniques may be
used to enhance hydrocarbon operations, such as hydrocarbon exploration
and hydrocarbon production. For example, the hydrocarbon exploration
operations involve any activity associated with determining the locations
of hydrocarbons in subsurface regions. Hydrocarbon exploration involves
activities conducted to obtain measurements through acquisitions of
measured data associated with the subsurface formation and the associated
modeling of the data to identify potential locations of hydrocarbon
accumulations. Accordingly, hydrocarbon exploration includes acquiring
measurement data, modeling of the measurement data to form subsurface
models and determining the likely locations for hydrocarbon reservoirs
within the subsurface. The measurement data may include seismic, gravity,
magnetic, electromagnetic and the like.

[0058] Further, hydrocarbon production operations involve any activity
associated with extracting hydrocarbons from a well or other opening.
Hydrocarbon production involve activities conducted to form the well
along with activities in or on the well after the well is completed.
Accordingly, hydrocarbon production or extraction includes not only
primary hydrocarbon extraction but also secondary and tertiary production
techniques, such as injection of gas or liquid for increasing drive
pressure, mobilizing the hydrocarbon or treating by, for example
chemicals or hydraulic fracturing the wellbore to promote increased flow,
well servicing, well logging, and other well and wellbore treatments.

[0059] The hydrocarbon operations are used to develop strategies. The
strategies may be used to explore for hydrocarbons and/or to produce
hydrocarbons. That is, based on the comparison, drilling of a well may be
performed to provide access to the hydrocarbon accumulation. Further, the
production may include installing or modifying a production facility for
the production of hydrocarbons from the production intervals that provide
access to the hydrocarbons in the subsurface formation. The production
facility may include one or more units to process and manage the flow of
production fluids, such as hydrocarbons and/or water, from the formation.
To access the production intervals, the production facility may be
coupled to a tree and various control valves via a control umbilical,
production tubing for passing fluids from the tree to the production
facility, control tubing for hydraulic or electrical devices, and a
control cable for communicating with other devices within the wellbore.
The strategy may adjust the well locations, fracture depths and patterns,
etc.

[0060] Beneficially, this method provides an enhancement in the production
and exploration of hydrocarbons. In particular, the method may be
utilized to enhance hydrocarbon exploration and hydrocarbon production
operations.

[0061] Persons skilled in the technical field will readily recognize that
in practical applications of the disclosed methodology, it is partially
performed on a computer, typically a suitably programmed digital
computer. Further, some portions of the detailed descriptions which
follow are presented in terms of procedures, steps, logic blocks,
processing and other symbolic representations of operations on data bits
within a computer memory. These descriptions and representations are the
means used by those skilled in the data processing arts to most
effectively convey the substance of their work to others skilled in the
art. In the present application, a procedure, step, logic block, process,
or the like, is conceived to be a self-consistent sequence of steps or
instructions leading to a desired result. The steps are those requiring
physical manipulations of physical quantities. Usually, although not
necessarily, these quantities take the form of electrical or magnetic
signals capable of being stored, transferred, combined, compared, and
otherwise manipulated in a computer system.

[0062] It should be borne in mind, however, that all of these and similar
terms are to be associated with the appropriate physical quantities and
are merely convenient labels applied to these quantities. Unless
specifically stated otherwise as apparent from the following discussions,
it is appreciated that throughout the present application, discussions
utilizing the terms such as "processing" or "computing", "calculating",
"comparing", "determining", "displaying", "copying," "producing,"
"storing," "adding," "applying," "executing," "maintaining," "updating,"
"creating," "constructing" "generating" or the like, refer to the action
and processes of a computer system, or similar electronic computing
device, that manipulates and transforms data represented as physical
(electronic) quantities within the computer system's registers and
memories into other data similarly represented as physical quantities
within the computer system memories or registers or other such
information storage, transmission or display devices.

[0063] Embodiments of the present techniques also relate to an apparatus
for performing the operations herein. This apparatus may be specially
constructed for the required purposes, or it may comprise a
general-purpose computer selectively activated or reconfigured by a
computer program stored in the computer (e.g., one or more sets of
instructions). Such a computer program may be stored in a computer
readable medium. A computer-readable medium includes any mechanism for
storing or transmitting information in a form readable by a machine
(e.g., a computer). For example, but not limited to, a computer-readable
(e.g., machine-readable) medium includes a machine (e.g., a computer)
readable storage medium (e.g., read only memory ("ROM"), random access
memory ("RAM"), magnetic disk storage media, optical storage media, flash
memory devices, etc.), and a machine (e.g., computer) readable
transmission medium (electrical, optical, acoustical or other form of
propagated signals (e.g., carrier waves, infrared signals, digital
signals, etc.)).

[0064] Furthermore, as will be apparent to one of ordinary skill in the
relevant art, the modules, features, attributes, methodologies, and other
aspects of the invention can be implemented as software, hardware,
firmware or any combination of the three. Of course, wherever a component
of the present invention is implemented as software, the component can be
implemented as a standalone program, as part of a larger program, as a
plurality of separate programs, as a statically or dynamically linked
library, as a kernel loadable module, as a device driver, and/or in every
and any other way known now or in the future to those of skill in the art
of computer programming Additionally, the present invention is in no way
limited to implementation in any specific operating system or
environment.

[0065] Further, one or more embodiments may include methods that are
performed by executing one or more sets of instructions to perform
modeling enhancements in various stages. For example, the method may
include executing one or more sets of instructions to perform comparisons
between thresholds current statuses or indications along with
transmitting data between modules, components and/or sensors.

[0066] As an example, a computer system may be utilized and configured to
implement on or more of the present aspects. The computer system may
include a processor; memory in communication with the processor; and a
set of instructions stored on the memory and accessible by the processor,
wherein the set of instructions, when executed, are configured to:
generate a mapping of a model mesh representing a physical domain of the
subsurface region, with faults, to an optimized mesh representing a
continuous design space in which all faults are removed, said mapping
being designed to minimize deformation in mesh cells; assign values of
the material property to continuous volumes in the optimized mesh to
generate a model of the material property in the design space, and using
that model to generate a model of the material property in the faulted
physical domain; and provide, store or display the model of the material
property in the faulted physical domain for hydrocarbon prospecting or
reservoir development in the subsurface region. Further, the set of
instructions for minimizing deformation in mesh cells may be configured
to: generate a tessellated mesh dividing the physical domain into cells,
and recording geometric quality of each cell; design stitching
constraints to stitch together discontinuities at fault boundaries, or
alternatively stitching discontinuities by node relocation, thereby
truncating cells at fault boundaries; and optimize the mesh in an
iterative optimization procedure, subject to the stitching constraints,
with the optimization aimed at minimizing degradation in geometric
quality from the recorded geometric quality due to the stitching
constraints, wherein all mesh nodes are free to move, or all mesh nodes
are free to move except mesh nodes associated with the stitching together
of discontinuities at fault boundaries, which mesh nodes are relocated to
an average position and held fixed there. Also, the set of instructions
may be configured to adjust locations of mesh nodes under influence of
the constraints; penalize worst quality cells based on a global grid
quality measure computed by adding together quality metrics computed on
cells in the mesh; compute quality metrics on every cell in the mesh are
based on combining a shape quality indicator with a size metric;
determine shape quality indicator based on a Jacobian of a mapping from a
unit square to a general quadrilateral cell; generate cells in the
generated tessellated mesh having edges that do not cross horizon or
fault surfaces; stitch boundary points on a surface of discontinuity;
parameterize the surface of discontinuity and stitching pairs of points
intermediate between said boundary points; stitch constraints are based
on minimizing distance between two points, on opposite sides of a fault
boundary, to be stitched together, said two points having been determined
to be co-located before the fault occurred; stitch constraints by
including, in a cost or objective function that is being minimized in the
optimization, a term containing an expression for said distance between
two points to be stitched together, expressed with a Lagrange multiplier
or as a penalty term and/or generate tessellated mesh is cut or
non-conforming across discontinuities.

[0067] As an example, FIG. 8 is a block diagram of a computer system 800
that may be used to perform any of the methods disclosed herein. A
central processing unit (CPU) 802 is coupled to system bus 804. The CPU
802 may be any general-purpose CPU, although other types of architectures
of CPU 802 (or other components of exemplary system 800) may be used as
long as CPU 802 (and other components of system 800) supports the present
techniques as described herein. The CPU 802 may execute the various
logical instructions according to disclosed aspects and methodologies.
For example, the CPU 802 may execute machine-level instructions for
performing processing according to aspects and methodologies disclosed
herein.

[0068] The computer system 800 may also include computer components such
as a random access memory (RAM) 806, which may be SRAM, DRAM, SDRAM, or
the like. The computer system 800 may also include read-only memory (ROM)
808, which may be PROM, EPROM, EEPROM, or the like. RAM 806 and ROM 808
hold user and system data and programs, as is known in the art. The
computer system 800 may also include an input/output (I/O) adapter 810, a
communications adapter 822, a user interface adapter 824, and a display
adapter 818. The I/O adapter 810, the user interface adapter 824, and/or
communications adapter 822 may, in certain aspects and techniques, enable
a user to interact with computer system 800 to input information.

[0069] The I/O adapter 810 preferably connects a storage device(s) 812,
such as one or more of hard drive, compact disc (CD) drive, floppy disk
drive, tape drive, etc. to computer system 800. The storage device(s) may
be used when RAM 806 is insufficient for the memory requirements
associated with storing data for operations of embodiments of the present
techniques. The data storage of the computer system 800 may be used for
storing information and/or other data used or generated as disclosed
herein. The communications adapter 822 may couple the computer system 800
to a network (not shown), which may enable information to be input to
and/or output from system 800 via the network (for example, a wide-area
network, a local-area network, a wireless network, any combination of the
foregoing). User interface adapter 824 couples user input devices, such
as a keyboard 828, a pointing device 826, and the like, to computer
system 800. The display adapter 818 is driven by the CPU 802 to control,
through a display driver 816, the display on a display device 820.
Information and/or representations of one or more 2D canvases and one or
more 3D windows may be displayed, according to disclosed aspects and
methodologies.

[0070] The architecture of system 800 may be varied as desired. For
example, any suitable processor-based device may be used, including
without limitation personal computers, laptop computers, computer
workstations, and multi-processor servers. Moreover, embodiments may be
implemented on application specific integrated circuits (ASICs) or very
large scale integrated (VLSI) circuits. In fact, persons of ordinary
skill in the art may use any number of suitable structures capable of
executing logical operations according to the embodiments.

[0071] In one or more embodiments, the method may be implemented in
machine-readable logic, such that a set of instructions or code that,
when executed, performs operations from memory.

[0072] It should be understood that the preceding is merely a detailed
description of specific embodiments of the invention and that numerous
changes, modifications, and alternatives to the disclosed embodiments can
be made in accordance with the disclosure here without departing from the
scope of the invention. The preceding description, therefore, is not
meant to limit the scope of the invention. Rather, the scope of the
invention is to be determined only by the appended claims and their
equivalents. It is also contemplated that structures and features
embodied in the present examples can be altered, rearranged, substituted,
deleted, duplicated, combined, or added to each other.

[0073] The foregoing description is directed to particular embodiments of
the present invention for the purpose of illustrating it. It will be
apparent, however, to one skilled in the art, that many modifications and
variations to the embodiments described herein are possible. All such
modifications and variations are intended to be within the scope of the
present invention, as defined by the appended claims.